The de Broglie nonlinear wave equation and The coaxial spacetime transistor

This manuscript reinstates Louis de Broglie’s foundational Law of Phase Harmony not as a historical postulate, but as the governing mechanical invariant of nanoscale solid-state transport. We demonstrate that the three-symbol matter-wave equation (λp = h) is not a standalone statistical approximation, but a direct geometric consequence of this law, enforcing absolute phase-locking between a localized energetic core and its extended spatial metric. By applying this rigid phase constraint to the closed polyhedral topology of a fullerene carbon lattice, we derive a zero-leakage, bi-directional coaxial quantum waveguide. The Law of Phase Harmony dictates that the counter-propagating surface vectors must maintain strict topological orthogonality, yielding a theoretical Coaxial Molecular Transistor that operates entirely free of continuous bandgap approximations or thermal wave-shear.